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Stat 155 Lecture Notes Game Theory Stat 155 Lecture Notes Game Theory Professor: Oscar Hernan Madrid Padilla Scribe: Daniel Raban Contents 1 Combinatorial Games5 1.1 Subtraction game and definitions........................5 1.2 Combinatorial games as graphs.........................6 1.3 Existence of a winning strategy.........................6 1.4 Chomp.......................................6 2 Nim and Rim8 2.1 Nim........................................8 2.2 Rim........................................9 3 Staircase Nim and Partisan Games 10 3.1 Staircase Nim................................... 10 3.2 Partisan Games.................................. 11 3.2.1 Partisan subtraction game........................ 11 3.2.2 Hex.................................... 11 4 Two Player Zero-Sum Games 13 4.1 Pick a Hand.................................... 13 4.2 Zero-sum games.................................. 13 4.3 Von-Neumann's minimax theorem....................... 15 5 Solving Two-player Zero-sum Games 16 5.1 Saddle points................................... 16 5.2 Removing dominated pure strategies...................... 17 5.3 2 × 2 games.................................... 18 1 6 Domination and the Principle of Indifference 19 6.1 Domination by multiple rows or columns.................... 19 6.2 The principle of indifference........................... 20 6.3 Using the principle of indifference........................ 20 7 Symmetry in Two Player Zero-sum Games 22 7.1 Submarine Salvo................................. 22 7.2 Invariant vectors and matrices.......................... 22 7.3 Using invariance to solve games......................... 23 8 Nash Equilibria, Linear Programming, and von Neumann's Minimax Theorem 25 8.1 Nash equilibria.................................. 25 8.1.1 Optimality of Nash equilibria...................... 25 8.1.2 Indifference and Nash Equilibria.................... 26 8.2 Solving zero-sum games using matrix inversion................ 26 8.3 Linear programming: an aside.......................... 27 8.4 Proof of von Neumann's minimax theorem................... 28 9 Gradient Ascent, Series Games, and Parallel Games 30 9.1 Gradient ascent.................................. 30 9.2 Series and parallel games............................ 31 9.2.1 Series games............................... 31 9.2.2 Parallel games.............................. 31 9.2.3 Electric networks............................. 32 10 Two Player General-Sum Games 33 10.1 General-sum games and Nash equilibria.................... 33 10.2 Examples of general-sum games......................... 34 11 Two-Player and Multiple-Player General-Sum Games 36 11.1 More about two-player general-sum games................... 36 11.1.1 Cheetahs and gazelles.......................... 36 11.1.2 Comparing two-player zero-sum and general-sum games....... 36 11.2 Multiplayer general-sum games......................... 37 12 Indifference of Nash Equilibria, Nash's Theorem, and Potential Games 40 12.1 Indifference of Nash equilibria in general-sum games............. 40 12.2 Nash's theorem.................................. 41 12.3 Potential and Congestion games......................... 42 12.3.1 Congestion games............................ 42 12.3.2 Potential games.............................. 43 2 13 Evolutionary Game Theory 44 13.1 Criticisms of Nash equilibria........................... 44 13.2 Evolutionarily stable strategies......................... 44 13.3 Examples of strategies within populations................... 45 14 Evolutionary Game Theory of Mixed Strategies and Multiple Players 47 14.1 Relationships between ESSs and Nash equilibria............... 47 14.2 Evolutionary stability against mixed strategies................ 47 14.3 Multiplayer evolutionarily stable strategies................... 49 15 Correlated Equilibria and Braess's Paradox 50 15.1 An example of inefficient Nash equilibria.................... 50 15.2 Correlated strategy pairs and equilibria.................... 50 15.3 Interpretations and comparisons to Nash equilibria.............. 52 15.4 Braess's paradox................................. 52 16 The Price of Anarchy 54 16.1 Flows and latency in networks.......................... 54 16.2 The price of anarchy for linear and affine latencies.............. 56 16.3 The impact of adding edges........................... 58 17 Pigou Networks and Cooperative Games 59 17.1 Pigou networks.................................. 59 17.2 Cooperative games................................ 60 18 Threat Strategies and Nash Bargaining in Cooperative Games 63 18.1 Threat strategies in games with transferable utility.............. 63 18.2 Nash bargaining model for nontransferable utility games........... 65 19 Models for Transferable Utility 67 19.1 Nash's bargaining theorem and relationship to transferable utility...... 67 19.2 Multiplayer transferable utility games..................... 68 19.2.1 Allocation functions and Gillies' core.................. 68 19.2.2 Shapley's axioms for allocation functions................ 70 20 Shapley Value 71 20.1 Shapley's axioms................................. 71 20.2 Junta games.................................... 71 20.3 Shapley's theorem................................ 72 3 21 Examples of Shapley Value and Mechanism Design 74 21.1 Examples of Shapley value............................ 74 21.2 Examples of mechanism design......................... 75 22 Voting 78 22.1 Voting preferences, preference communication, and Borda count....... 78 22.2 Properties of voting systems........................... 79 22.3 Violating IIA and Arrow's Impossibility theorem............... 80 23 Impossibility Theorems and Properties of Voting Systems 82 23.1 The Gibbard-Satterthwaite theorem...................... 82 23.2 Properties of voting systems........................... 83 23.3 Positional voting rules.............................. 84 4 1 Combinatorial Games 1.1 Subtraction game and definitions Consider a \subtraction game" with 2 players and 15 chips. Players alternate moves, with player 1 starting. At each move, the player can remove 1 or 2 chips. A player wins when they take the last chip (so the other player cannot move). Let x be the number of chips remaining. Suppose you move next. Can you guarantee a win? Let's look at a few examples. If x 2 f1; 2g, the player who moves can take the remaining chip(s) and win. If x = 3, the second player has the advantage; no matter what player 1 does, player 2 will be presented with 1 or 2 chips. Write N as the set of positions where the next player to move can guarantee a win, provided they play optimally. Write P as the set of positions where the other player, the player that moved previously, can guarantee a win, provided that they play optimally. So 0; 3 2 P , 1; 2 2 N. In the case of our original game, 15 2 P . Definition 1.1. A combinatorial game is a game with two players (players 1 and 2) and a set X of positions. For each player, there is a set of legal moves between positions, M1;M2 ⊆ X × X (current position, next position). Players alternately choose moves, starting from some starting position x0, and play continues until some player cannot move. The game has a winner or loser and follows normal or misere play. Definition 1.2. In a combinatorial game, normal play means that the player who cannot move loses the game. Definition 1.3. In a combinatorial game, misere play means that the player who cannot move wins the game. Definition 1.4. An impartial game has the same set of legal moves for both players; i.e. M1 = M2.A partisan game has different sets of legal moves for the players. Definition 1.5. A terminal position for a player is a position in which the player has no legal move to another position; i.e. x is terminal for player i if there is no y 2 X with (x; y) 2 Mi. Definition 1.6. A combinatorial grame is progressively bounded if, for every starting po- sition x0 2 X, there is a finite bound on the number of moves before the game ends. Definition 1.7. A strategy for a player is a function that assigns a legal move to each non-terminal position. If XNT is the set of non-terminal positions for player i, then Si : XNT ! X is a strategy for player i if, for all x 2 XNT ,(x; Si(x)) 2 Mi. Definition 1.8. A winning strategy for a player from position x is a strategy that is guaranteed to a result in a win for that player. 5 Example 1.1. The subtraction game is an impartial combinatorial game. The positions are X = f0; 1; 2;:::; 15g, and the moves are f(x; y 2 X × X : y 2 fx − 1; x − 2gg. The terminal position for both players is 0. The game is played using normal play. It is progressively bounded because from x 2 X, there can be no more than x moves until the terminal position. A winning strategy for any starting position x 2 N is S(x) = 3bx=3c. 1.2 Combinatorial games as graphs Impartial combinatorial games can be thought of as directed graphs. Think of the positions as nodes and the moves as directed edges between the nodes Terminal positions are nodes without outgoing edges. Example 1.2. What does the graph look like for the subtraction game? Every edge from a node in P leads into a node in N. There is also an edge from every node in N to a node in P . The winning strategy chooses one of these edges. Acyclic graphs correspond to progressively bounded games. B(x) is the maximum length along the graph from node x to a terminal position. 1.3 Existence of a winning strategy Theorem 1.1. In a progressively bounded, impartial combinatorial game, X = N [ P . That is, from any initial position, one of the players has a winning strategy. Proof. By definition, N; P
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